Answer :
To find which ratios are equivalent to [tex]\(6:8\)[/tex], we need to simplify [tex]\(6:8\)[/tex] and compare it with the other given ratios.
1. Simplify [tex]\(6:8\)[/tex]:
The greatest common divisor (GCD) of 6 and 8 is 2. So, we can simplify the ratio:
[tex]\[
\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}
\][/tex]
2. Check each given ratio:
- 16 to 24:
Simplify [tex]\(\frac{16}{24}\)[/tex].
The GCD of 16 and 24 is 8. So,
[tex]\[
\frac{16}{24} = \frac{16 \div 8}{24 \div 8} = \frac{2}{3} \quad (\text{not equal to } \frac{3}{4})
\][/tex]
- [tex]\(\frac{15}{20}\)[/tex]:
Simplify [tex]\(\frac{15}{20}\)[/tex].
The GCD of 15 and 20 is 5. So,
[tex]\[
\frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4} \quad (\text{equal to } \frac{3}{4})
\][/tex]
- [tex]\(\frac{36}{40}\)[/tex]:
Simplify [tex]\(\frac{36}{40}\)[/tex].
The GCD of 36 and 40 is 4. So,
[tex]\[
\frac{36}{40} = \frac{36 \div 4}{40 \div 4} = \frac{9}{10} \quad (\text{not equal to } \frac{3}{4})
\][/tex]
- 24:36:
Simplify [tex]\(\frac{24}{36}\)[/tex].
The GCD of 24 and 36 is 12. So,
[tex]\[
\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3} \quad (\text{not equal to } \frac{3}{4})
\][/tex]
- 9:12:
Simplify [tex]\(\frac{9}{12}\)[/tex].
The GCD of 9 and 12 is 3. So,
[tex]\[
\frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4} \quad (\text{equal to } \frac{3}{4})
\][/tex]
- 12 to 16:
Simplify [tex]\(\frac{12}{16}\)[/tex].
The GCD of 12 and 16 is 4. So,
[tex]\[
\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4} \quad (\text{equal to } \frac{3}{4})
\][/tex]
Based on the simplifications, the ratios that are equivalent to [tex]\(6:8\)[/tex] (or [tex]\(\frac{3}{4}\)[/tex]) are:
- [tex]\(\frac{15}{20}\)[/tex]
- [tex]\(9:12\)[/tex]
- [tex]\(12 \text{ to } 16\)[/tex]
1. Simplify [tex]\(6:8\)[/tex]:
The greatest common divisor (GCD) of 6 and 8 is 2. So, we can simplify the ratio:
[tex]\[
\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}
\][/tex]
2. Check each given ratio:
- 16 to 24:
Simplify [tex]\(\frac{16}{24}\)[/tex].
The GCD of 16 and 24 is 8. So,
[tex]\[
\frac{16}{24} = \frac{16 \div 8}{24 \div 8} = \frac{2}{3} \quad (\text{not equal to } \frac{3}{4})
\][/tex]
- [tex]\(\frac{15}{20}\)[/tex]:
Simplify [tex]\(\frac{15}{20}\)[/tex].
The GCD of 15 and 20 is 5. So,
[tex]\[
\frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4} \quad (\text{equal to } \frac{3}{4})
\][/tex]
- [tex]\(\frac{36}{40}\)[/tex]:
Simplify [tex]\(\frac{36}{40}\)[/tex].
The GCD of 36 and 40 is 4. So,
[tex]\[
\frac{36}{40} = \frac{36 \div 4}{40 \div 4} = \frac{9}{10} \quad (\text{not equal to } \frac{3}{4})
\][/tex]
- 24:36:
Simplify [tex]\(\frac{24}{36}\)[/tex].
The GCD of 24 and 36 is 12. So,
[tex]\[
\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3} \quad (\text{not equal to } \frac{3}{4})
\][/tex]
- 9:12:
Simplify [tex]\(\frac{9}{12}\)[/tex].
The GCD of 9 and 12 is 3. So,
[tex]\[
\frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4} \quad (\text{equal to } \frac{3}{4})
\][/tex]
- 12 to 16:
Simplify [tex]\(\frac{12}{16}\)[/tex].
The GCD of 12 and 16 is 4. So,
[tex]\[
\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4} \quad (\text{equal to } \frac{3}{4})
\][/tex]
Based on the simplifications, the ratios that are equivalent to [tex]\(6:8\)[/tex] (or [tex]\(\frac{3}{4}\)[/tex]) are:
- [tex]\(\frac{15}{20}\)[/tex]
- [tex]\(9:12\)[/tex]
- [tex]\(12 \text{ to } 16\)[/tex]
